An introduction to central simple algebras and their applications to wireless communication (Q2841741)

The book under review contains a presentation of the theory of (associative finite-dimensional) central simple algebras (abbr, CSA-s), and a systematic introduction to its applications in communication theory. It is addressed primarily to graduate students in mathematics. Whenever possible, the authors draw the two main lines of the discussed material, following a parallel approach. The main content is presented in nine chapters, supplemented by three appendices: on the basic theory of tensor products of vector spaces and algebras over a field; a glimpse of algebraic number theory; generalities on hermitian lattices and complex ideal lattices. NEWLINENEWLINEChapter I includes preliminaries on algebras over a field \(K\), basic definitions and examples concerning CSA-s, and an introduction to space-time coding. Chapter II introduces the reader to quaternion \(K\)-algebras, where \(\text{char}(K)\neq 2\), and to codes based on such algebras. The Frobenius theorem on finite-dimensional division algebras over the field \(\mathbb R\) of real numbers is also included. In Chapter III, the authors show the stability of CSA-s under tensor products and under scalar extensions. They also prove the Skolem-Noether Theorem, Wedderburn's Structure Theorem and the Centralizer Theorem. NEWLINENEWLINEThese theorems are used in Chapter IV together with standard material on splitting fields for proving Wedderburn's Theorem about the commutativity of finite division rings. One is also introduced to the reduced characteristic polynomial of an element of a CSA, to the concept and basic properties of the reduced norm, and as an application, to the minimum determinant of a space-time code (an ''equivalent'' of the minimum Hamming distance). NEWLINENEWLINEIn Chapter V the authors define Brauer equivalence of CSA-s, introduce the Brauer group \(\text{Br}(K)\) and prove that it is an abelian torsion group. This is used to define the exponent \(\exp(A)\) of a CSA, say \(A\), to deduce Brauer's relations between \(\exp(A)\) and the (Schur) index \(\text{ind}(A)\), and to establish the existence of a primary tensor product decomposition for \(A\). NEWLINENEWLINEIn Chapter VI, they introduce the notion of a crossed product, deduce Noether's Multiplication Theorem for crossed products, and prove that CSA-s are Brauer equivalent to such products. The developed theory is used for constructing families of codes. NEWLINENEWLINEChapter VII concentrates on cyclic algebras, the simplest type of crossed products. It presents their main properties over arbitrary fields, and includes an overview, without proofs, of the theory of CSA-s over local fields and number fields. Specifically, one can find explicit criteria to decide whether such an algebra is a division one, which are used to design codes based on cyclic division algebras. NEWLINENEWLINEIn Chapter VIII the authors consider central simple \(K\)-algebras of degree \(4\). They show that these algebras are crossed products over a biquadratic extension of \(K\) and fully describe them by generators and relations. When \(K\) is a number field, they find a decision criteria similar to the one in Chapter VII, and apply it to code constructions. NEWLINENEWLINEChapter IX deals with the problem of finding whether a CSA \(B/K\) is endowed with a unitary involution (as usual, \(K\) is supposed to be a quadratic extension of a field \(K _ 0\) with \(\text{char}(K _ 0) \neq 2\)); a particular attention is paid to the special case where \(B\) is a crossed product. It is explained how CSA-s with unitary involution can be used in coding theory via construction of unitary codes.NEWLINENEWLINENEWLINENEWLINEThe book is well-written, the material is illustrated by various examples and exercises, and its presentation gives a concrete flavour of the discussed topics. Besides to graduate students in mathematics, it can be useful to mathematicians and mathematically sophisticated coding theorists.